Questions — Edexcel C12 (247 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel C12 2018 June Q1
  1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 1 } { \sqrt { ( x + 1 ) } }\), with the values
    for \(y\) rounded to 3 decimal places where necessary.
\(x\)03691215
\(y\)10.50.3780.3160.277
  1. Complete the table by giving the value of \(y\) corresponding to \(x = 15\)
  2. Use the trapezium rule with all the values of \(y\) from the completed table to find an approximate value for $$\int _ { 0 } ^ { 15 } \frac { 1 } { \sqrt { ( x + 1 ) } } \mathrm { d } x$$ giving your answer to 2 decimal places.
Edexcel C12 2018 June Q2
2. $$f ( x ) = a x ^ { 3 } + 2 x ^ { 2 } + b x - 3$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ) the remainder is 1
  1. Show that $$a + 4 b = 28$$ When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 17
  2. Find the value of \(a\) and the value of \(b\).
Edexcel C12 2018 June Q3
3. The line \(l _ { 1 }\) passes through the points \(A ( - 1,4 )\) and \(B ( 5 , - 8 )\)
  1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is perpendicular to the line \(l _ { 1 }\) and passes through the point \(B ( 5 , - 8 )\)
  2. Find an equation for \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers.
    II
    "
Edexcel C12 2018 June Q4
4. Given that $$y = \frac { 64 x ^ { 6 } } { 25 } , x > 0$$ express each of the following in the form \(k x ^ { n }\) where \(k\) and \(n\) are constants.
  1. \(y ^ { - \frac { 1 } { 2 } }\)
  2. \(( 25 y ) ^ { \frac { 2 } { 3 } }\)
Edexcel C12 2018 June Q5
  1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 1 + \frac { x } { 3 } \right) ^ { 18 }$$ giving each term in its simplest form.
(b) Use the answer to part (a) to find an estimated value for \(\left( \frac { 31 } { 30 } \right) ^ { 18 }\), stating the value of \(x\) that you have used and showing your working. Give your estimate to 4 decimal places. II
Edexcel C12 2018 June Q6
6. Find the exact values of \(x\) for which $$2 \log _ { 5 } ( x + 5 ) - \log _ { 5 } ( 2 x + 2 ) = 2$$ Give your answers as simplified surds.
Edexcel C12 2018 June Q7
7. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = u _ { n } - 5 , \quad n \geqslant 1 \end{aligned}$$ Find the values of
  1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
  2. \(u _ { 100 }\)
  3. \(\sum _ { i = 1 } ^ { 100 } u _ { i }\)
Edexcel C12 2018 June Q8
8. The equation \(( k - 4 ) x ^ { 2 } - 4 x + k - 2 = 0\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$k ^ { 2 } - 6 k + 4 > 0$$
  2. Find the exact range of possible values for \(k\).
Edexcel C12 2018 June Q9
9. A cyclist aims to travel a total of 1200 km over a number of days. He cycles 12 km on day 1
He increases the distance that he cycles each day by \(6 \%\) of the distance cycled on the previous day, until he reaches the total of 1200 km .
  1. Show that on day 8 he cycles approximately 18 km . He reaches his total of 1200 km on day \(N\), where \(N\) is a positive integer.
  2. Find the value of \(N\). The cyclist stops when he reaches 1200 km .
  3. Find the distance that he cycles on day \(N\). Give your answer to the nearest km .
Edexcel C12 2018 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce06b37a-aa57-4256-bec8-7277c8a9fc65-24_348_593_221_534} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Diagram not drawn to scale Figure 1 shows a semicircle with centre \(O\) and radius \(3 \mathrm {~cm} . X Y\) is the diameter of this semicircle. The point Z is on the circumference such that angle \(X O Z = 1.3\) radians. The shaded region enclosed by the chord \(X Z\), the arc \(Z Y\) and the diameter \(X Y\) is a template for a badge. Find, giving each answer to 3 significant figures,
  1. the length of the chord \(X Z\),
  2. the perimeter of the template \(X Z Y X\),
  3. the area of the template.
Edexcel C12 2018 June Q11
11. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where $$f ^ { \prime } ( x ) = \frac { 5 x ^ { 2 } + 4 } { 2 \sqrt { x } } - 5$$ It is given that the point \(P ( 4,14 )\) lies on \(C\).
  1. Find \(\mathrm { f } ( x )\), writing each term in a simplified form.
  2. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C12 2018 June Q12
12. [In this question solutions based entirely on graphical or numerical methods are not acceptable.]
  1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), $$5 \sin \left( x + 65 ^ { \circ } \right) + 2 = 0$$ giving your answers in degrees to one decimal place.
  2. Find, for \(0 \leqslant \theta < 2 \pi\), all the solutions of $$12 \sin ^ { 2 } \theta + \cos \theta = 6$$ giving your answers in radians to 3 significant figures.
Edexcel C12 2018 June Q13
13. The point \(A ( 9 , - 13 )\) lies on a circle \(C\) with centre the origin and radius \(r\).
  1. Find the exact value of \(r\).
  2. Find an equation of the circle \(C\). A straight line through point \(A\) has equation \(2 y + 3 x = k\), where \(k\) is a constant.
  3. Find the value of \(k\). This straight line cuts the circle again at the point \(B\).
  4. Find the exact coordinates of point \(B\).
Edexcel C12 2018 June Q14
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce06b37a-aa57-4256-bec8-7277c8a9fc65-40_611_1214_219_548} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ^ { 2 } ( 2 x + 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(\left( - \frac { 1 } { 2 } , 0 \right)\), touches it at \(( 2,0 )\) and crosses the \(y\)-axis at ( 0,4 ). There is a maximum turning point at the point marked \(P\).
  1. Use \(\mathrm { f } ^ { \prime } ( x )\) to find the exact coordinates of the turning point \(P\). A second curve \(C _ { 2 }\) has equation \(y = \mathrm { f } ( x + 1 )\).
  2. Write down an equation of the curve \(C _ { 2 }\) You may leave your equation in a factorised form.
  3. Use your answer to part (b) to find the coordinates of the point where the curve \(C _ { 2 }\) meets the \(y\)-axis.
  4. Write down the coordinates of the two turning points on the curve \(C _ { 2 }\)
  5. Sketch the curve \(C _ { 2 }\), with equation \(y = \mathrm { f } ( x + 1 )\), giving the coordinates of the points where the curve crosses or touches the \(x\)-axis.
Edexcel C12 2018 June Q15
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce06b37a-aa57-4256-bec8-7277c8a9fc65-44_851_1506_212_260} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A design for a logo consists of two finite regions \(R _ { 1 }\) and \(R _ { 2 }\), shown shaded in Figure 3 .
The region \(R _ { 1 }\) is bounded by the straight line \(l\) and the curve \(C\).
The region \(R _ { 2 }\) is bounded by the straight line \(l\), the curve \(C\) and the line with equation \(x = 5\)
The line \(l\) has equation \(y = 8 x + 38\)
The curve \(C\) has equation \(y = 4 x ^ { 2 } + 6\)
Given that the line \(l\) meets the curve \(C\) at the points \(( - 2,22 )\) and \(( 4,70 )\), use integration to find
  1. the area of the larger lower region, labelled \(R _ { 1 }\)
  2. the exact value of the total area of the two shaded regions. Given that $$\frac { \text { Area of } R _ { 1 } } { \text { Area of } R _ { 2 } } = k$$
  3. find the value of \(k\).
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Edexcel C12 2019 June Q1
  1. The 4th term of a geometric series is 125 and the 7th term is 8
    1. Show that the common ratio of this series is \(\frac { 2 } { 5 }\)
    2. Hence find, to 3 decimal places, the difference between the sum to infinity and the sum of the first 10 terms of this series.
Edexcel C12 2019 June Q2
  1. Find the value of \(a\) and the value of \(b\) for which \(\frac { 8 ^ { x } } { 2 ^ { x - 1 } } \equiv 2 ^ { a x + b }\)
  2. Hence solve the equation \(\frac { 8 ^ { x } } { 2 ^ { x - 1 } } = 2 \sqrt { 2 }\)
Edexcel C12 2019 June Q4
4. Given that $$y = 5 x ^ { 2 } + \frac { 1 } { 2 x } + \frac { 2 x ^ { 4 } - 8 } { 5 \sqrt { x } } \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form.
(6)
HAVI SIHI NI JINM ION OCVIIV SIHI NI JINAM ION OAVIUV SIHI NI JIIIM ION OC
Edexcel C12 2019 June Q5
5. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$\begin{aligned} u _ { 1 } & = 1
u _ { n + 1 } & = k - \frac { 8 } { u _ { n } } \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Write down expressions for \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\). Given that \(u _ { 3 } = 6\)
  2. find the possible values of \(k\).
Edexcel C12 2019 June Q6
6. (a) Find, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), the binomial expansion of $$\left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$ giving each term in its simplest form.
(b) Hence find the coefficient of \(x\) in the expansion of $$\left( 3 + \frac { 2 } { x } \right) ^ { 2 } \left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$
Edexcel C12 2019 June Q7
7. (a) Sketch the graph of \(y = \sin \left( x + \frac { \pi } { 6 } \right) , \quad 0 \leqslant x \leqslant 2 \pi\) Show the coordinates of the points where the graph crosses the \(x\)-axis. The table below gives corresponding values of \(x\) and \(y\) for \(y = \sin \left( x + \frac { \pi } { 6 } \right)\).
The values of \(y\) are rounded to 3 decimal places where necessary.
\(x\)0\(\frac { \pi } { 8 }\)\(\frac { \pi } { 4 }\)\(\frac { 3 \pi } { 8 }\)\(\frac { \pi } { 2 }\)
\(y\)0.50.7930.9660.9910.866
(b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin \left( x + \frac { \pi } { 6 } \right) \mathrm { d } x$$ Give your answer to 2 decimal places.
Edexcel C12 2019 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de511cb3-35c7-4225-b459-a136b6304b78-20_547_463_269_735} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale Figure 2 shows the design for a company logo. The design consists of a triangle \(A B E\) joined to a sector \(B C D E\) of a circle with radius 6 cm and centre \(E\). The line \(A E\) is perpendicular to the line \(D E\) and the length of \(A E\) is 9 cm . The size of angle \(D E B\) is 3.5 radians, as shown in Figure 2.
  1. Find the length of the arc BCD. Find, to one decimal place,
  2. the perimeter of the logo,
  3. the area of the logo.
Edexcel C12 2019 June Q9
9. \(\mathrm { f } ( x ) = ( x + k ) \left( 3 x ^ { 2 } + 4 x - 16 \right) + 32 , \quad\) where \(k\) is a constant (a) Write down the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + k )\). When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is 15
(b) Show that \(k = 2\)
(c) Hence factorise \(\mathrm { f } ( x )\) completely. \section*{9.} " .
\(\mathrm { f } ( x ) = ( x + k ) \left( 3 x ^ { 2 } + 4 x - 16 \right) + 32 , \quad\) where \(k\) is a constant
Edexcel C12 2019 June Q10
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } + 4 x + p y + 123 = 0$$ where \(p\) is a constant. Given that the point \(( 1,16 )\) lies on \(C\),
  1. find
    1. the value of \(p\),
    2. the coordinates of the centre of \(C\),
    3. the radius of \(C\).
  2. Find an equation of the tangent to \(C\) at the point ( 1,16 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-31_33_19_2668_1896}
Edexcel C12 2019 June Q11
11. The straight line \(l\) has equation \(y = m x - 2\), where \(m\) is a constant. The curve \(C\) has equation \(y = 2 x ^ { 2 } + x + 6\) The line \(l\) does not cross or touch the curve \(C\).
  1. Show that \(m\) satisfies the inequality $$m ^ { 2 } - 2 m - 63 < 0$$
  2. Hence find the range of possible values of \(m\).